1. **Problem Statement:** Find the unique fixed probability vector $\mathbf{p}$ of the stochastic matrix $$AB=\begin{bmatrix}1 & 2 & 1 \\ 2 & 1 & 3 \\ 1 & 4 & 1 \\ 6 & 1 & 3 \\ 1 & 4 & 1 \\ 3 & 1 & 3\end{bmatrix}.$$ 2. **Understanding the problem:** A fixed probability vector $\mathbf{p}$ for a stochastic matrix $M$ satisfies $$M\mathbf{p} = \mathbf{p}$$ and $$\sum_i p_i = 1,$$ with all $p_i \geq 0$. This means $\mathbf{p}$ is an eigenvector of $M$ corresponding to eigenvalue 1, normalized to sum to 1. 3. **Check matrix dimensions and stochastic property:** The given matrix $AB$ is $6 \times 3$, which is not square. A stochastic matrix must be square with each column or row summing to 1. Since $AB$ is not square, it cannot have a unique fixed probability vector in the usual sense. 4. **Conclusion:** The problem as stated is inconsistent because $AB$ is not a square stochastic matrix. To find a fixed probability vector, the matrix must be square and stochastic (rows or columns sum to 1). Please verify the matrix or provide a square stochastic matrix for the fixed vector calculation.